Differentiation is easy!

Calculus Level 4

What is the first derivative of f ( x ) = ∑ n = 1 ∞ sin ⁡ ( π × n 2 × x ) π × n 2 ? f(x) = \sum\limits_{n = 1}^\infty \frac{\sin(\pi\times n^2\times x)}{\pi\times n^2}?

A) e sin ⁡ ( π 2 6 x ) \text{A) } e^{\sin(\frac{\pi^2}{6}x)}

B) e tan ⁡ x 2 \text{B) } e^{\tan{x^2}}

C) ln ⁡ [ sin ⁡ ( π 2 6 ) cos ⁡ ( e sec ⁡ − 1 ( x 2 ) ] \text{C) } \ln[\sin(\frac{\pi^2}{6})\cos(e^{\sec^{-1}(x^2)}]

D) cos ⁡ − 1 ( x x × π 2 ) \text{D) } \cos^{-1}(x^x \times \pi^2)

E) ∑ n = 1 ∞ sin ⁡ ( π × n 2 × x ) π × n 2 \text{E) } \sum\limits_{n = 1}^\infty \frac{\sin(\pi\times n^2\times x)}{\pi\times n^2}

F) None of the above \text{F) None of the above}

B D F E C A

Caleb Townsend by Caleb Townsend , 26, USA

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1 solution

Caleb Townsend
Mar 7, 2015

In fact, the function is continuous everywhere, but differentiable nowhere (that is, the first derivative exists nowhere). Since all of the other options exist at least somewhere, none of them can be the derivative. Therefore, the answer is F) None of the above \boxed{\text{F) None of the above}}

For more information on this function, check here .

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